Reaction quotient

In chemical thermodynamics, the reaction quotient (Q_r or just Q) ^[1] is a dimensionless quantity that provides a measurement of the relative amounts of products and reactants present in a reaction mixture for a reaction with well-defined overall stoichiometry at a particular point in time. Mathematically, it is defined as the ratio of the activities (or molar concentrations) of the product species over those of the reactant species involved in the chemical reaction, taking stoichiometric coefficients of the reaction into account as exponents of the concentrations. In equilibrium, the reaction quotient is constant over time and is equal to the equilibrium constant.

A general chemical reaction in which α moles of a reactant A and β moles of a reactant B react to give ρ moles of a product R and σ moles of a product S can be written as

${\displaystyle {\ce {\it {\alpha \,{\rm {A{}+{\it {\beta \,{\rm {B{}<=>{\it {\rho \,{\rm {R{}+{\it {\sigma \,{\rm {S{}}}}}}}}}}}}}}}}}}}$.

The reaction is written as an equilibrium even though, in many cases, it may appear that all of the reactants on one side have been converted to the other side. When any initial mixture of A, B, R, and S is made, and the reaction is allowed to proceed (either in the forward or reverse direction), the reaction quotient Q_r, as a function of time t, is defined as ^[2]

${\displaystyle Q_{\text{r}}(t)={\frac {\{\mathrm {R} \}_{t}^{\rho }\{\mathrm {S} \}_{t}^{\sigma }}{\{\mathrm {A} \}_{t}^{\alpha }\{\mathrm {B} \}_{t}^{\beta }}},}$

where {X}_t denotes the instantaneous activity ^[3] of a species X at time t. A compact general definition is

${\displaystyle Q_{\text{r}}(t)=\prod _{j}[a_{j}(t)]^{\nu _{j}},}$

where П_j denotes the product across all j-indexed variables, a_j(t) is the activity of species j at time t, and ν_j is the stoichiometric number (the stoichiometric coefficient multiplied by +1 for products and –1 for starting materials).

Relationship to K (the equilibrium constant)

As the reaction proceeds with the passage of time, the species' activities, and hence the reaction quotient, change in a way that reduces the free energy of the chemical system. The direction of the change is governed by the Gibbs free energy of reaction by the relation

${\displaystyle \Delta _{\mathrm {r} }G=RT\ln(Q_{\mathrm {r} }/K)}$,

where K is a constant independent of initial composition, known as the equilibrium constant. The reaction proceeds in the forward direction (towards larger values of Q_r) when Δ_rG < 0 or in the reverse direction (towards smaller values of Q_r) when Δ_rG > 0. Eventually, as the reaction mixture reaches chemical equilibrium, the activities of the components (and thus the reaction quotient) approach constant values. The equilibrium constant is defined to be the asymptotic value approached by the reaction quotient:

${\displaystyle Q_{\mathrm {r} }\to K}$ and ${\displaystyle \Delta _{\mathrm {r} }G\to 0\quad (t\to \infty )}$.

The timescale of this process depends on the rate constants of the forward and reverse reactions. In principle, equilibrium is approached asymptotically at t → ∞; in practice, equilibrium is considered to be reached, in a practical sense, when concentrations of the equilibrating species no longer change perceptibly with respect to the analytical instruments and methods used.

If a reaction mixture is initialized with all components having an activity of unity, that is, in their standard states, then

${\displaystyle Q_{\mathrm {r} }=1}$ and ${\displaystyle \Delta _{\mathrm {r} }G=\Delta _{\mathrm {r} }G^{\circ }=-RT\ln K\quad (t=0)}$.

This quantity, Δ_rG°, is called the standard Gibbs free energy of reaction. ^[4]

All reactions, regardless of how favorable, are equilibrium processes, though practically speaking, if no starting material is detected after a certain point by a particular analytical technique in question, the reaction is said to go to completion.

In biochemistry

In biochemistry, the reaction quotient is often referred to as the mass-action ratio with the symbol ${\displaystyle \Gamma }$.

Example

The burning of octane, C₈H₁₈ + ²⁵/₂ O₂ → 8CO₂ + 9H₂O has a Δ_rG° ~ –240 kcal/mol, corresponding to an equilibrium constant of 10¹⁷⁵, a number so large that it is of no practical significance, since there are only ~5 × 10²⁴ molecules in a kilogram of octane.

References

1. Cohen, E Richard; Cvitas, Tom; Frey, Jeremy G; Holström, Bertil; Kuchitsu, Kozo; Marquardt, Roberto; Mills, Ian; Pavese, Franco; Quack, Martin, eds. (2007). Quantities, Units and Symbols in Physical Chemistry (3 ed.). Cambridge: Royal Society of Chemistry. doi:10.1039/9781847557889. ISBN   978-0-85404-433-7.
2. Zumdahl, Steven; Zumdahl, Susan (2003). Chemistry (6th ed.). Houghton Mifflin. ISBN   0-618-22158-1.
3. Under certain circumstances (see chemical equilibrium) each activity term such as {A} may be replaced by a concentration term, [A]. Both the reaction quotient and the equilibrium constant are then concentration quotients.
4. The standard free energy of reaction can be determined using the difference between the sum of the standard free energies of formation of products and the sum of the standard free energies of formation of reactants, accounting for stoichiometries: ${\textstyle \Delta _{\mathrm {r} }G^{\circ }=\sum _{\mathrm {prod.} }^{i}\nu _{i}\Delta _{\mathrm {f} }G^{\circ }-\sum _{\mathrm {react.} }^{j}\nu _{j}\Delta _{\mathrm {f} }G^{\circ }}$.

External links

Reaction quotient tutorials

• tutorial I ^[1] No longer accessible as of November 2023
• tutorial II ^[2]
• tutorial III ^[3]

1. "elchem website tutorial 1". Archived from the original on 23 January 2020. Retrieved 28 April 2021.
2. "elchem website tutorial 2". Archived from the original on 23 January 2020. Retrieved 28 April 2021.
3. "elchem website tutorial 3". Archived from the original on 23 January 2020. Retrieved 28 April 2021.